# Parallel Iterative Methods for Nonsymmetric Large-Scale Problems.

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## Abstract

Much of the supercomputer research so far has concentrated on implementations of iterative methods for symmetric positive definite matrices, in particular the preconditioned conjugate gradient (PCG) algorithm, and an understanding of the parallel issues of this algorithm is emerging. Much less work has been done regarding iterative methods for nonsymmetric problems. In this report, parallel implementations of some important algorithms for nonsymmetric systems of equations, namely, the quasi-minimal residual (QMR) and transpose-free QMR (TFQMR) algorithms for solving sparse systems of equations, and the Lanczos bidiagonalization method for treating least squares problems with large condition numbers are investigated. Thus, a large amount of applications that lead to nonsymmetric problems is covered. The research is motivated by applications in soil pollution and helioseismology. The implementations were done on two modern computer architectures: a PARAGON XPS 10 with 140 nodes located at the Research Centre Juelich GmbH (KFA), and a Convex SPP Exemplar with 8 processors located at UNI-C. The developed parallel algorithms show an advantageous scaling behavior on these target machines.